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ACT Math : How to subtract exponents

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Example Questions

Example Question #1 : How To Subtract Exponents

Reduce $\frac {x^9y^5}{x^3y^2z}$ to simplest form.

Possible Answers:

$\frac {x^6y^3}{z}$

x^3y^3z

$\frac {x^3y^3}{z}$

x^6y^3z

$\frac {x^3}{y^3z}$

Correct answer:

$\frac {x^6y^3}{z}$

Explanation:

When dividing terms with the same bases but different exponents, you will need to subtract all the pertinent exponents.

$\frac {x^9}{x^3}$ becomes x^6 ,

$\frac {y^5}{y^2}$ becomes y^3 ,

and $\frac {1}{z}$ stays the same because there is no other z term to combine with it.

Thus resulting in:

$\frac{x^6y^3}{z}$

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Example Question #1 : How To Subtract Exponents

Simplify: 3² * (4²³ - 4²¹)

Possible Answers:

None of the other answers

3^3 * 4^21

3^3 * 4^21 * 5

3^21

4^4

Correct answer:

3^3 * 4^21 * 5

Explanation:

Begin by noting that the group (4²³ - 4²¹) has a common factor, namely 4²¹. You can treat this like any other constant or variable and factor it out. That would give you: 4²¹(4² - 1). Therefore, we know that:

3² * (4²³ - 4²¹) = 3² * 4²¹(4² - 1)

Now, 4² - 1 = 16 - 1 = 15 = 5 * 3. Replace that in the original:

3² * 4²¹(4² - 1) = 3² * 4²¹(3 * 5)

Combining multiples withe same base, you get:

3³ * 4²¹ * 5

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Example Question #1 : How To Subtract Exponents

Simplify. Leave no negative exponents in the final answer.

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3}$

Possible Answers:

None of these are the correct answer.

$\frac{y^5}{z^{5}}$

$\frac{y^3}{z^5}$

$a^{6}y^{5}z^{5}$

$-\frac{y^{5}}{z^{5}}$

Correct answer:

$\frac{y^5}{z^{5}}$

Explanation:

The first step in the problem is to combine like terms in the numerator, remembering that $a^{m}\cdot a^{n} = a^{m+n}$ :

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3}$

Next, we resolve the numerator, using $\frac{a^{m}}{a^{n}}=a^{m-n}$ and $a^{0} =1$ :

$\frac{a^{3}y^{7}z^{-2}}{a^{3}y^{2}z^3} = y^{5}z^{-5}$

Lastly, simplify the negative exponent using $a^{-m}=\frac{1}{a^{m}}$

$y^{5}z^{-5} = \frac{y^{5}}{z^{5}}$

Thus,

$\frac{a^{3}y^{3}y^{4}z^{-2}}{a^{3}y^{2}z^3} = \frac{y^{5}}{z^{5}}$

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Example Question #1 : How To Subtract Exponents

Simplify to remove fractions:

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}}$

Possible Answers:

$w^{-2}x^{9}y^{2}z$

None of these are correct.

$w^{-3}y^{2}z$

$w^{3}x^{3}y^{2}z^3$

$w^{2}y^{-4}z^{-1}$

Correct answer:

$w^{-3}y^{2}z$

Explanation:

The first step is to simplify each fraction by dividing like terms, remembering that $\frac{a^{m}}{a^{n}}=a^{m-n}$ , to get:

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = (w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5})$

Next, combine using multiplication and the rule $a^{m}\cdot a^{n}= a^{m+n}$ :

$(w^{-1}y^{2}z^{-4})\cdot (w^{-2}z^{5}) = w^{-3}y^{2}z$

Since the problem specifies that we must avoid fractions, we will not eliminate the negative exponents.

So,

$\frac{wx^{3}y^{2}z^{-3}}{w^{2}x^{3}z}\cdot \frac{w^{-2}x^{3}}{x^{3}z^{-5}} = w^{-3}y^{2}z$

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Example Question #2 : How To Subtract Exponents

Simplify the following: $\frac{x^{10}}{x^{6}}$

Possible Answers:

$x^{-4}$

$x^{16}$

$x^{4}$

$x^{\frac{10}{6}}$

Correct answer:

$x^{4}$

Explanation:

When dividing exponential expressions with the same base, subtract the exponents. In this problem, the exponents are and . When subtracted, the result is . Thus, the correct answer is $x^{4}$ .

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Example Question #1 : How To Subtract Exponents

$\frac{10^{6}}{10^{2}}$ can be written as which of the following?

A. $10^{\frac{6}{2}}$

B. $10^{4}$

C. 10000

Possible Answers:

B and C

C only

B only

A, B and C

A and C

Correct answer:

B and C

Explanation:

A is not equivalent because exponents in denominators mean subtraction of exponents and not division of them. Furthermore, A, when computed, comes out to 1000 instead of 10000 .

B is equivalent by the aforementioned exponential property, while C is simply computing the expression.

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Example Question #1 : How To Subtract Exponents

Simplify: $\frac{x^{3}}{x}$

Possible Answers:

$x^{3}$

$x^{2}$

Correct answer:

$x^{2}$

Explanation:

When two exponents with the same base are being divided, subtract the exponent of the denominator from the exponent of the numerator to yield a new exponent. Attach that exponent to the base, and that is your answer.

In this case, subtract from . That yields as the new exponent and $x^{2}$ as the answer.

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ACT Math : How to subtract exponents

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #2 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

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